Attached is a screenshot from the latest major. There is a numerical value presented below one of the clan flags (the side varies). My question is what does the number represent?
It is the fan level (as the Wiki calls it) of the team.
I found a few Reddits asking about this. This one states:
[The] Amount of pennants level equipped of spectators in Dota TV. Basically every lvl 1 pennant adds 1 to the number, and lvl 2 pennants adds 2 to the number.
This Reddit says the same thing basically (with reference to Na'Vi):
It's the number of spectators in the DotA2 client who have a Na'Vi team pennant.
The Dota 2 Wiki calls this number the Fan Level on the pennants page:
Pennants are equippable items that allow users to show support for a team during tournaments, by increasing a team's Fan Level - visible in-game to other DotaTV users.
Those are pennants: They used to be sold on the Dota Store, and it was a way to cheer for your team, there are different levels to existing pennants and the level just multiplies how much a pennant is worth (pennant lvl 1 adds 1).
Iirc you can not buy pennants anymore, and even if you can, there are no pennants for new-ish teams.